A ball is dropped from a height of H meters and after each bounce it rebounds...

 

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A ball is dropped from a height of H meters and after each bounce it rebounds to a fraction p of its previous height. The time required for the ball to come to rest (after an infinite number of bounces) is given by T(p) = 2H (1+), where g = 9.8 m/s² is the acceleration due to gravity. This model neglects the effects of air resistance. We will consider dropping a ball from a height of H = 4.9 m, so that=1. Thus T(p) = 1+ √P = 1. Within the context of this problem, what is the domain of this function? 2. Use a graphing utility to graph the function T(p) for relevant values of p. 3. If the ball rebounds to p = 0.5 of its previous height after each bounce, how long does it take to come to a rest? 4. Evaluate and interpret T(0). Is this the value you expected? Explain. Evaluate and interpret T(p), as p approaches 1. Is this the value you expect? Explain. A cylindrical tank with a height of H m and a radius of R m is filled with water. At time t = 0, a circular drain in the bottom of the tank with a radius of r m is opened and water is allowed to flow out of the tank. A good model (based on Torricelli's Law) predicts that the height of water in the tank after t seconds is given by the function T(p) = [√H - √28 (5) ² 1] ², for ≥ 0, where g ≈ 9.8 m/s² is the acceleration due to gravity. Before assigning specific values to r, R, and H, let's do some preliminary analysis. 2 2 A ball is dropped from a height of H meters and after each bounce it rebounds to a fraction p of its previous height. The time required for the ball to come to rest (after an infinite number of bounces) is given by T(p) = 2H (1+), where g = 9.8 m/s² is the acceleration due to gravity. This model neglects the effects of air resistance. We will consider dropping a ball from a height of H = 4.9 m, so that=1. Thus T(p) = 1+ √P = 1. Within the context of this problem, what is the domain of this function? 2. Use a graphing utility to graph the function T(p) for relevant values of p. 3. If the ball rebounds to p = 0.5 of its previous height after each bounce, how long does it take to come to a rest? 4. Evaluate and interpret T(0). Is this the value you expected? Explain. Evaluate and interpret T(p), as p approaches 1. Is this the value you expect? Explain. A cylindrical tank with a height of H m and a radius of R m is filled with water. At time t = 0, a circular drain in the bottom of the tank with a radius of r m is opened and water is allowed to flow out of the tank. A good model (based on Torricelli's Law) predicts that the height of water in the tank after t seconds is given by the function T(p) = [√H - √28 (5) ² 1] ², for ≥ 0, where g ≈ 9.8 m/s² is the acceleration due to gravity. Before assigning specific values to r, R, and H, let's do some preliminary analysis. 2 2

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